See Example 5 JAR Paper #2 (Page 7) submitted:

"A synthetic proof without the order relation of the unordered version of Example 5
is a very good testing exercise for full understanding of unordered geometries discussed in this section."

The proof is thus diagram-independent taking care of all three cases, which can been seen in this animation, with the single proof text.

I will write a check of $50 for first 4 persons who send me a correct single diagram-independent proof (text) to

chou@cs.wichita.edu